The molecular bound states of atomic quantum gases near a Feshbach resonance were analyzed. It was observed that a renormalizable field-theoretic model for Feshbach coupling has exact solutions for the eigenstate in the low-density two-particle sector. It was shown that the proposed model has the capability to describe a variety of cases with both positive and negative background scattering length. The solutions interpolating between BEC and BCS theories provides a more general variational ansatz for resonant superfluidity and related problems.